Linear Programming – Simplex Method

1. Linear Programming – Simplex Method

2. Linear Programming - Review • Graphical Method: • What is the feasible region? • Where was optimal solution found? • What is primary limitation of graphical method? • Conversion to Standard Form: • - • - • -

3. Linear Programming – Review • Solving Systems of Linear Equations: • What is a basic solution? • How did we obtain a basic solution? • What is a basic feasible solution? • Relationship between graphical and algebraic • representation of the feasible region: • corner point basic solution

4. Linear Programming – Review Fundamental insight – the optimal solution to a linear program, if it exists, is also a basic feasible solution. Naïve approach – solve for all basic solutions and find the feasible solution with the largest value (maximization problem). What is the problem with this approach? – there are possible basic solutions, where m is the number of constraints and n is the number of variables.

5. Linear Programming – Simplex Algorithm Step 1 Convert the LP to standard form. Step 2 Obtain a bfs (if possible) from the standard form. Step 3 Determine whether the current bfs is optimal. Step 4 If the current bfs is not optimal, then determine which nonbasic basic variable should become a basic variable and which basic variable should become a nonbasic variable to find a new bfs with a better objective function value. (pivot operation) Step 5 Use EROs to find the new bfs with the better objective function value. Go back to step 3. Operations Research, Wayne L. Winston

6. Linear Programming – Simplex Method Review Simplex Handouts

7. Linear Programming – Simplex Method Minimization Problems: Min Z = cx  (-) Max Z = -cx Ex. Min 2x1 – 3x2 + x3 s.t. x1 + 2x2 < 5 2x1 - 3x3 > 10 x1, x2, x3 > 0 (-)Max -2x1 + 3x2 - x3 s.t. x1 + 2x2 < 5 2x1 - 3x3 > 10 x1, x2, x3 > 0

8. Linear Programming – Simplex Method: Computational Problems Breaking Ties in Selection of Non-Basic Variable – if tie for non-basic variable with largest relative profit ( ), arbitrarily select incoming variable. Ties in Minimum Ratio Rule (Degeneracy) – if more than one basic variable have same minimum ratio, select either variable to leave the basis. This will result in a basic variable taking on a value of 0. When this occurs, the solution is referred to as a degenerate basic feasible solution. When this occurs, you may transition through more than one simplex tableau with the same objective (Z) value.

9. Linear Programming – Simplex Method: Computational Problems Unbounded Solutions – if when performing the minimum ratio rule, none of the ratios are positive, then the solution is unbounded (e.g Max Z = or Min = - ).

10. Simplex Method – Finding an Initial Basic Feasible Solution Min Z = -3x1 + x2 + x3 s.t. x1 – 2x2 + x3 <= 11 -4x1 + x2 +2x3 >= 3 2x1 - x3 = -1 x1, x2, x3 >= 0 Standard Form: (-) Max Z = 3x1 - x2 - x3 s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 = 3 -2x1 + x3 = 1 x1, x2, x3, x4, x5 >= 0

11. Simplex Method – Finding an Initial Basic Feasible Solution (-) Max Z = 3x1 - x2 - x3 s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 = 3 -2x1 + x3 = 1 x1, x2, x3, x4, x5 >= 0 Only x4 is basic. Introduce artificial variables. s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 + x6 = 3 -2x1 + x3 + x7 = 1 x1, x2, x3, x4, x5, x6, x7 >= 0

12. Simplex Method – Solve Using Big-M Method Let M be an arbitrarily large number, then: (-) Max Z = 3x1 - x2 - x3 + 0x4 + 0x5 – Mx6 – Mx7 s.t. x1 – 2x2 + x3 + x4 = 11 -4x1 + x2 +2x3 -x5 + x6 = 3 -2x1 + x3 + x7 = 1 x1, x2, x3, x4, x5, x6, x7 >= 0 Note: If the simplex algorithm terminates with one of the artificial variables as a basic variable, then the original problem has no feasible solution.